$\beta _{ij}$ (Spatially non-stationary spatial model) with ARD prior

$$\beta_{ij}\vert.\sim N(B_{\beta_{ij}}/A_{\beta_{ij}},1/A_{\beta_{ij}})$$ (34)

where:

$$A_{\beta_{ij}} = \sum_{t}(\phi_{\epsilon_i}q_{j(t-1)}^2+\phi_{\epsilon_j}q_{i(t-1)}^2) +\phi_{\beta_{ij}}$$

$$B_{\beta_{ij}} = \sum_{t} \left[\phi_{\epsilon_i}q_{j(t-1)} \left( q_{it}-\sum_{p=1}^P \alpha_{ip} q_{i(t-p)} - \right. \right.$$

$$\left. \left. \sum_{k\in{\cal N}_i:k\neq j} \beta_{ik} q_{k(t-1)} \right) + \phi_{\epsilon_j}q_{i(t-1)} \right.$$

$$\left. \left( q_{jt}-\sum_{p=1}^P \alpha_{jp} q_{j(t-p)} \right. \right.$$

$$\left. \left. -\sum_{k\in{\cal N}_j:k\neq i} \beta_{jk} q_{k(t-1)} \right) \right]$$ (35)